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\begin{document}
\title{A zoo of ordinals}
\author{David A. Madore}
\maketitle
{\footnotesize
\immediate\write18{sh ./vc > vcline.tex}
Git: \input{vcline.tex}
\immediate\write18{echo ' (stale)' >> vcline.tex}
\par}
\textbf{\textcolor{red}{Preliminary version:}} the labels in this
document are subject to change.
\section{Recursive ordinals}
\setcounter{comcnt}{-1}
\ordinal $0$ (zero). This is the smallest ordinal, and the only one that is
neither successor nor limit.
\ordinal $1$ (one). This is the smallest successor ordinal.
\ordinal $2$.
\ordinal $42$ (cf. \cite[chapter 27]{Adams1981}).
\ordinal $\omega$. This is the smallest limit ordinal, and the
smallest infinite ordinal.
\ordinal $\omega+1$. This is the smallest infinite successor ordinal.
\ordinal $\omega2$.
\ordinal $\omega^2$.
\ordinal $\omega^\omega$.
\ordinal $\omega^{\omega^\omega}$.
\ordinal $\varepsilon_0 = \varphi(1,0)$. This is the limit of
$\omega, \omega^\omega, \omega^{\omega^\omega}, \ldots$, smallest
fixed point of $\xi \mapsto \omega^\xi$; in general, $\alpha \mapsto
\varepsilon_\alpha = \varphi(1,\alpha)$ is defined as the function
enumerating the fixed points of $\xi \mapsto \omega^\xi$. This is the
proof-theoretic ordinal of Peano arithmetic.
\ordinal $\varepsilon_1 = \varphi(1,1)$.
\ordinal $\varepsilon_\omega$.
\ordinal $\varepsilon_{\varepsilon_0}$.
\ordinal $\varphi(2,0)$. This is the limit of $\varepsilon_0,
\varepsilon_{\varepsilon_0}, \ldots$, smallest fixed point of $\xi
\mapsto \varepsilon_\xi$; in general, $\alpha \mapsto
\varphi(\gamma+1,\alpha)$ is defined as the function enumerating the
fixed points of $\xi \mapsto \varphi(\gamma,\xi)$.
\ordinal $\varphi(\omega,0)$. This is the smallest ordinal $>\omega$
closed under primitive recursive ordinal functions
(\cite[corollary 4.5]{Avigad2002}).
\ordinal The Feferman-Schütte ordinal $\Gamma_0 = \varphi(1,0,0)$
(also $\psi(\Omega^{\Omega})$ for an appropriate collapsing
function $\psi$). This is the limit of $\varepsilon_0, \penalty0
\varphi(\varepsilon_0,0), \penalty0 \varphi(\varphi(\varepsilon_0)),
\ldots$, smallest fixed point of $\xi \mapsto \varphi(\xi, 0)$. This
is the proof-theoretic ordinal of $\mathsf{ATR}_0$.
\ordinal The Ackermann ordinal $\varphi(1,0,0,0)$ (also
$\psi(\Omega^{\Omega^2})$ for an appropriate collapsing
function $\psi$).
\ordinal The “small” Veblen ordinal ($\psi(\Omega^{\Omega^\omega})$ for
an appropriate collapsing function $\psi$). This is the limit of
$\varphi(1,0), \penalty0 \varphi(1,0,0), \penalty0 \varphi(1,0,0,0),
\ldots$, the range of the Veblen functions with finitely many
variables.
\ordinal The “large” Veblen ordinal ($\psi(\Omega^{\Omega^\Omega})$
for an appropriate collapsing function $\psi$). This is the range of
the Veblen functions with up to that many variables.
\ordinal The Bachmann-Howard ordinal ($\psi(\varepsilon_{\Omega+1})$
for an appropriate collapsing function $\psi$). This is the
proof-theoretic ordinal of Kripke-Platek set theory ($\mathsf{KP}$).
\ordinal The countable collapse of $\varepsilon_{\Omega_\omega + 1}$
(“Takeuti-Feferman-Buchholz ordinal”), which is the proof-theoretic
ordinal of $\Pi^1_1$-comprehension + transfinite induction.
\ordinal\label{CollapseInaccessible} The countable collapse of
$\varepsilon_{I+1}$ where $I$ is the first inaccessible
(= $\Pi^1_0$-indescribable) cardinal. This is the proof-theoretic
ordinal of Kripke-Platek set theory augmented by the recursive
inaccessibility of the class of ordinals ($\mathsf{KPi}$), or, on the
arithmetical side, of $\Delta^1_2$-comprehension + transfinite
induction. See \cite{JaegerPohlers1983}.
(Compare •\ref{RecursivelyInaccessible}.)
\ordinal\label{CollapseMahlo} The countable collapse of
$\varepsilon_{M+1}$ where $M$ is the first Mahlo cardinal. This is
the proof-theoretic ordinal of $\mathsf{KPM}$.
See \cite{Rathjen1990}. (Compare •\ref{RecursivelyMahlo}.)
\ordinal\label{CollapseWeaklyCompact} The countable collapse of
$\varepsilon_{K+1}$ where $K$ is the first weakly compact
(= $\Pi^1_1$-indescribable) cardinal. This is the proof-theoretic
ordinal of $\mathsf{KP} + \Pi_3\hyphen\mathsf{Ref}$.
See \cite{Rathjen1994}. (Compare •\ref{RecursivelyWeaklyCompact}.)
\ordinal\label{CollapsePiTwoZeroIndescribable} The countable collapse
of $\varepsilon_{\Xi+1}$ where $\Xi$ is the first
$\Pi^2_0$-indescribable cardinal. This is the proof-theoretic ordinal
of $\mathsf{KP} + \Pi_\omega\hyphen\mathsf{Ref}$.
See \cite[part I]{Stegert2010} (in whose notation this ordinal would
be called $\Psi^{\varepsilon_{\Xi+1}}_{\mathbb{X}}$ where $\mathbb{X}
= (\omega^+; \mathsf{P}_0; \epsilon; \epsilon; 0)$).
(Compare •\ref{WeaklyStable}.)
\ordinal The proof-theoretic ordinal of $\mathsf{Stability}$:
see \cite[part II]{Stegert2010} (in whose notation this ordinal would
be called $\Psi^{\varepsilon_{\Upsilon+1}}_{\mathbb{X}}$ where
$\mathbb{X} = (\omega^+; \mathsf{P}_0; \epsilon; \epsilon; 0)$).
%
\section{Recursively large countable ordinals}
\ordinal\label{ChurchKleene} The Church-Kleene ordinal
$\omega_1^{\mathrm{CK}}$: the smallest admissible ordinal $>\omega$.
This is the smallest ordinal which is not the order type of a
recursive (equivalently: hyperarithmetic) well-ordering on $\omega$.
The $\omega_1^{\mathrm{CK}}$-recursive
(resp. $\omega_1^{\mathrm{CK}}$-semi-recursive) subsets of $\omega$
are exactly the $\Delta^1_1$ (=hyperarithmetic) (resp. $\Pi^1_1$)
subsets of $\omega$, and they are also exactly the subsets recursive
(resp. semi-recursive) in $\mathsf{E}$ (or $\mathsf{E}^\#$, \CHECKTHIS
[this is stated vaguely and without proof in \cite[§2, introductory
remarks]{HinmanMoschovakis1971}, and also alluded to, but with an
argument, in \cite[chapter VI, introductory remarks to §6 on
p. 316]{Hinman1978}; but the essential argument should be Gandy's
selection theorem, \cite[chapter VI, theorem 4.1 on
p. 292 or its corollary 4.3 on p. 294]{Hinman1978}]).
\ordinal $\omega_\omega^{\mathrm{CK}}$: the smallest limit of
admissibles. This ordinal is not admissible. This is the smallest
$\alpha$ such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of
$\Pi^1_1$-comprehension.
\ordinal\label{RecursivelyInaccessible} The smallest recursively
inaccessible ordinal: this is the smallest ordinal which is admissible
and limit of admissibles. This is the smallest ordinal $\alpha$ such
that $L_\alpha \models \mathsf{KPi}$, or, on the arithmetical side,
such that $L_\alpha \cap \mathscr{P}(\omega)$ is a model of
$\Delta^1_2$-comprehension. (Compare •\ref{CollapseInaccessible}.)
This is the smallest ordinal $\omega_1^{\mathsf{E}_1}$ not the order
type of a well-ordering recursive in the Tugué
functional $\mathsf{E}_1$ (\cite[chapter VIII, theorem 6.6 on
p. 421]{Hinman1978}), or equivalently, recursive in the hyperjump;
and for this $\alpha$ the $\alpha$-recursive
(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly the
subsets recursive (resp. semi-recursive) in $\mathsf{E}_1$
(\cite[chapter VIII, corollary 4.16 on p. 412]{Hinman1978}).
This is the smallest $\alpha$ such that $L_\alpha \models
\mathsf{KP}+\mathit{Beta}$, where $\mathit{Beta}$ asserts the
existence of a transitive collapse for any well-founded relation, or
equivalently, the smallest admissible $\alpha$ such that any ordering
which $L_\alpha$ thinks is a well-ordering is, indeed, a
well-ordering: see \cite[theorem 6.1 on p. 291]{Nadel1973}
(compare \cite{Harrison1968} for the negative result concerning the
ordinal $\omega_1^{\mathrm{CK}}$ of •\ref{ChurchKleene}; compare also
\cite{Gostanian1979} and •\ref{SigmaOneOne} for related facts).
\ordinal The smallest recursively hyperinaccessible ordinal: i.e., the
smallest recursively inaccessible which is a limit of recursively
inaccessibles.
\ordinal\label{RecursivelyMahlo} The smallest recursively Mahlo
ordinal: i.e., the smallest admissible ordinal $\alpha$ such that for
any $\alpha$-recursive function $f \colon \alpha \to \alpha$ there is
an admissible $\beta<\alpha$ which is closed under $f$. This is the
smallest ordinal $\alpha$ such that $L_\alpha \models \mathsf{KPM}$.
(Compare •\ref{CollapseMahlo}.)
This is the smallest ordinal not the order type of a well-ordering
recursive in the superjump (\cite{AczelHinman1974} and
\cite{Harrington1974}); and for this $\alpha$ the $\alpha$-recursive
(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly the
subsets recursive in the superjump (resp. semirecursive in the partial
normalization of the superjump, \cite[theorem 5 on
p. 50]{Harrington1974}).
Also note concerning this ordinal: \cite[corollary 9.4(ii) on
p. 348]{RichterAczel1974}.
\ordinal\label{RecursivelyWeaklyCompact} The smallest
$\Pi_3$-reflecting (``recursively weakly compact'') ordinal. This can
also be described as the smallest ``$2$-admissible'' ordinal:
see \cite[theorem 1.16 on p. 312]{RichterAczel1974}.
(Compare •\ref{CollapseWeaklyCompact}.)
Also the sup of the closure ordinals for $\Sigma_3$ inductive
operators: \cite[theorem A on p. 303]{RichterAczel1974}. For this
$\alpha$ the $\alpha$-semi-recursive subsets of $\omega$ are exactly
the $\Sigma_3$-inductively definable subsets of $\omega$
(\cite[theorem A on p. 303 and theorem D on p. 304]{RichterAczel1974};
see also \cite[example 4.12 on p. 370]{Simpson1978}).
\ordinal\label{WeaklyStable} The smallest $(+1)$-stable ordinal, i.e.,
the smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1}
L_{\alpha+1}$. This is the smallest $\Pi^1_0$-reflecting (i.e.,
$\Pi_n$-reflecting for every $n\in\omega$) ordinal: \cite[theorem 1.18
on p. 313 and 333]{RichterAczel1974}.
(Compare •\ref{CollapsePiTwoZeroIndescribable}.)
\ordinal\label{PiOneOne} The smallest $(^+)$-stable ordinal, i.e., the
smallest $\alpha$ such that $L_\alpha \mathrel{\preceq_1}
L_{\alpha^+}$ where $\alpha^+$ is the smallest admissible
ordinal $>\alpha$. This is the smallest $\Pi^1_1$-reflecting ordinal:
\cite[theorem 1.19 on p. 313 and 336]{RichterAczel1974}. Also the sup
of the closure ordinals for $\Pi^1_1$ inductive operators:
\cite[theorem B on p. 303 or 10.7 on p. 355]{RichterAczel1974} and
\cite[theorem A on p. 222]{Cenzer1974}. For this $\alpha$ the
$\alpha$-semi-recursive subsets of $\omega$ are exactly the
$\Pi^1_1$-inductively definable subsets of $\omega$ (\cite[theorem D
on p. 304]{RichterAczel1974}; see also \cite[example 4.13 on
p. 370]{Simpson1978}).
This is the smallest ordinal $\omega_1^{\mathsf{G}_1^\#}$ not the
order type of a well-ordering recursive in the nondeterministic
functional $\mathsf{G}_1^\#$ defined by $\mathsf{G}_1^\#(f) \approx
\{f(0)\}_{(\omega_1^f)^+}(f(1))$; and for this $\alpha$ the
$\alpha$-recursive (resp. $\alpha$-semi-recursive) subsets of $\omega$
are exactly the subsets recursive (resp. semi-recursive) in
$\mathsf{G}_1^\#$ (\cite[theorem 7.4 on p. 238]{Cenzer1974}).
\ordinal\label{SigmaOneOne} The smallest $\Sigma^1_1$-reflecting
ordinal. Also the sup of the closure ordinals for $\Sigma^1_1$
inductive operators: \cite[theorem B on p. 303 or 10.7 on
p. 355]{RichterAczel1974}. For this $\alpha$ the
$\alpha$-semi-recursive subsets of $\omega$ are exactly the
$\Sigma^1_1$-inductively definable subsets of $\omega$
(\cite[theorem D on p. 304]{RichterAczel1974}; see
also \cite[example 4.14 on p. 370]{Simpson1978}).
That this ordinal is greater than that of •\ref{PiOneOne}:
\cite[corollary 1 to theorem 6 on p.213]{Aanderaa1974}; also see:
\cite[theorem 6.5]{Simpson1978} and \cite{GostanianHrbacek1979}.
This is the smallest ordinal $\omega_1^{\mathsf{E}_1^\#}$ not the
order type of a well-ordering recursive in the nondeterministic
version $\mathsf{E}_1^\#$ of the Tugué functional $\mathsf{E}_1$; and
for this $\alpha$ the $\alpha$-recursive
(resp. $\alpha$-semi-recursive) subsets of $\omega$ are exactly the
subsets recursive (resp. semi-recursive) in $\mathsf{E}_1^\#$ (combine
\cite[theorem 1 on p. 313, theorem 2 on p. 318]{Aczel1970} and
\cite[theorem D on p. 304]{RichterAczel1974}).
This is the smallest admissible $\alpha$ which is not Gandy, i.e.,
such that every $\alpha$-recursive linear ordering of $\alpha$ of
which $L_{\alpha^+}$ thinks that it is a well-ordering (with
$\alpha^+$ being the next admissible) is, indeed, a well-ordering: see
\cite[theorem 6.6 on p. 377]{Simpson1978} and
\cite[theorem 3.3]{Gostanian1979} (on the terminology ``Gandy
ordinal'', see \cite{AbramsonSacks1976}: in \cite{Gostanian1979} the
same ordinals are called ``good'').
[\FINDTHIS: how stable is this ordinal?]
\ordinal The smallest $(^{++})$-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_{\alpha^{++}}$
where $\alpha^+,\alpha^{++}$ are the two smallest admissible
ordinals $>\alpha$. This is $\Sigma^1_1$-reflecting and greater than
the ordinal of •\ref{SigmaOneOne} (\cite[theorem 6.4 on
p. 376]{Simpson1978} and
proposition \ref{PlusPlusStableOrdinalIsSigmaOneOneReflecting} below).
\ordinal The smallest inaccessibly-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta$ where
$\beta$ is the smallest recursively inaccessible
(cf. •\ref{RecursivelyInaccessible}) ordinal $>\alpha$.
\ordinal The smallest Mahlo-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta$ where
$\beta$ is the smallest recursively Mahlo
(cf. •\ref{RecursivelyMahlo}) ordinal $>\alpha$.
\ordinal The smallest doubly $(+1)$-stable ordinal, i.e., the smallest
$\alpha$ such that $L_\alpha \mathrel{\preceq_1} L_\beta
\mathrel{\preceq_1} L_{\beta+1}$ (cf. •\ref{WeaklyStable}).
\ordinal\label{NonprojectibleStable} The smallest stable ordinal under
a nonprojectible ordinal, i.e., the smallest $\alpha$ such that
$L_\alpha \mathrel{\preceq_1} L_\beta$ where $\beta$ is the smallest
nonprojectible (the ordinal of •\ref{Nonprojectible}).
This is the smallest ordinal $\omega_1^{\mathbf{R}}$ not the order
type of a well-ordering recursive in a certain type $3$ functional
$\mathbf{R}$ defined in \cite{Harrington1975}; and for this $\alpha$
the $\alpha$-recursive subsets of $\omega$ are exactly the subsets
recursive in $\mathbf{R}$. (See also \cite{John1986} and
\cite[example 4.10 on p. 369]{Simpson1978}.)
\ordinal\label{Nonprojectible} The smallest nonprojectible ordinal,
i.e., the smallest $\beta$ such that $\beta$ is a limit of
$\beta$-stable ordinals (ordinals $\alpha$ such that $L_\alpha
\mathrel{\preceq_1} L_\beta$ (cf. •\ref{NonprojectibleStable}); in
other words, the smallest $\beta$ such that $L_\beta \models
\mathsf{KPi}+$“the stable ordinals are unbounded”. This is the
smallest ordinal $\beta$ such that $L_\beta \models
\mathsf{KP}\omega+\Sigma_1\hyphen\textsf{Sep}$ (cf. \cite[chapter V,
theorem 6.3 on p. 175]{Barwise1975}), or such that $L_\beta \cap
\mathscr{P}(\omega)$ is a model of $\Pi^1_2$-comprehension
(cf. \cite[theorem VII.3.24 on p. 267 and theorem VII.5.17 on
p. 292]{Simpson2009}).
In Jensen's terminology (\cite{Jensen1972}), this is the smallest
ordinal $\beta$ such that $\rho_1^\beta > \omega$, and in fact the
smallest $\beta>\omega$ such that $\rho_1^\beta = \beta$: that is, the
smallest ordinal $\beta$ such that every $\Sigma_1(L_\beta)$ subset
of $\omega$ is $\beta$-finite. Sometimes also called the smallest
“strongly admissible” (or “strongly $\Sigma_1$-admissible”) ordinal.
\ordinal The smallest (weakly) $\Sigma_2$-admissible ordinal. This is
the smallest ordinal $\beta$ such that $L_\beta \models
\mathsf{KP}\omega+\Delta_2\hyphen\textsf{Sep}$, or such that $L_\beta
\cap \mathscr{P}(\omega)$ is a model of $\Delta^1_3$-comprehension
(cf. \cite[theorem VII.3.24 on p. 267 and theorem VII.5.17 on
p. 292]{Simpson2009}).
In Jensen's terminology (\cite{Jensen1972}), this is the smallest
ordinal $\beta$ such that $\eta_2^\beta > \omega$, and in fact the
smallest $\beta>\omega$ such that $\eta_2^\beta = \beta$: that is, the
smallest ordinal $\beta$ such that every $\Delta_2(L_\beta)$ subset
of $\omega$ is $\beta$-finite.
In the terminology of \cite[appendix]{MarekSrebrny1973}, this is the
first $\Delta_2$-gap ordinal.
\ordinal The ordinal of ramified analysis (often written $\beta_0$).
This is the smallest $\beta$ such that $L_\beta \models \bigwedge_n
\Sigma_n\hyphen\textsf{Sep}$ (the full separation scheme), or such
that $L_\beta \cap \mathscr{P}(\omega)$ is a model of full
second-order analysis (second-order comprehension), and in fact
$L_\beta \models \mathsf{ZFC}^-$ (that is, $\mathsf{ZFC}$ minus the
powerset axiom).
This starts the first gap in the constructible universe, and this gap
is of length $1$: see \cite{Putnam1963} and \cite[corollary 4.5 on
p. 374]{MarekSrebrny1973}.
Note that this ordinal is $(+1)$-stable (cf. •\ref{WeaklyStable}) but
not $(+2)$-stable: \cite[corollary to theorem 6.14 on
p. 384]{MarekSrebrny1973}.
\ordinal The start of the first gap of length $2$ in the constructible
universe. If $\beta$ is this ordinal then $\beta$ is the $\beta$-th
gap ordinal (\cite[theorem 4.17 on p. 377]{MarekSrebrny1973}).
\ordinal The first ordinal $\beta$ which starts a gap of
length $\beta$ in the constructible universe.
\ordinal\label{OmegaOneSmallestModelKPWithOmegaOne} The ordinal $\beta
= \omega_1^{L_\alpha}$ where $\alpha$ is ordinal
of •\ref{SmallestModelKPWithOmegaOne}. Then by construction $\beta$
starts a gap of length $\alpha = \beta^+$ (the next admissible
ordinal).
\ordinal\label{SmallestModelKPWithOmegaOne} The smallest ordinal
$\alpha$ such that $L_\alpha \models \mathsf{KP}+$“$\omega_1$ exists”,
i.e., the smallest admissible $\alpha$ which is not locally countable,
or equivalently, the smallest $\alpha$ such that $L_\alpha \models
\mathsf{KP}+$“$\mathscr{P}(\omega)$ exists”
(cf. proposition \ref{KPExistenceOfOmegaOneImpliesExistenceOfPOmega}).
\ordinal The smallest ordinal $\alpha$ such that $L_\alpha \models
\mathsf{ZFC}^-+$“$\omega_1$ exists”, or equivalently such that
$L_\alpha \models \mathsf{ZFC}^-+$“$\mathscr{P}(\omega)$ exists”
(cf. proposition \ref{KPExistenceOfOmegaOneImpliesExistenceOfPOmega}).
This is the start of the first third-order gap (\cite[theorem 3.7 on
p. 372]{MarekSrebrny1973}) in the constructible universe.
%
%
%
\ordinal\label{OmegaOneSmallestModelZFC} The smallest uncountable
ordinal $\omega_1^{L_\alpha}$ in the smallest model $L_{\alpha}$
of $\mathsf{ZFC}$, assuming it exists (see •\ref{SmallestModelZFC}).
This ordinal is $\alpha$-stable.
\ordinal\label{SmallestModelZFC} The smallest ordinal $\alpha$ such
that $L_\alpha \models \mathsf{ZFC}$ (assuming it exists), i.e., the
height of the minimal model of $\mathsf{ZFC}$.
\ordinal\label{Stable} The smallest stable ordinal $\sigma$, i.e., the
smallest $\sigma$ such that $L_\sigma \mathrel{\preceq_1} L$, or
equivalently $L_\sigma \mathrel{\preceq_1} L_{\omega_1}$. The set
$L_\sigma$ is the set of all $x$ which are $\Sigma_1$-definable in $L$
without parameters (\cite[chapter V, corollary 7.9(i) on
p. 182]{Barwise1975}).
This ordinal is projectible to $\omega$ (i.e., in Jensen's
terminology), $\rho_1^\sigma = \omega$ (\cite[chapter V,
theorem 7.10(i) on p. 183]{Barwise1975}).
This is the smallest ordinal $\delta^1_2$ which not the order type of
a well-ordering $\Delta^1_2$ on $\omega$; and in fact, for this
$\sigma = \delta^1_2$, the $\sigma$-recursive
(resp. $\sigma$-semi-recursive) subsets of $\omega$ are exactly the
$\Delta^1_2$ (resp. $\Sigma^1_2$) subsets of $\omega$
(\cite[chapter V, theorem 8.2 on p. 189 and corollary 8.3 on
p. 191]{Barwise1975}).
This is also the smallest $\Sigma^1_2$-reflecting ordinal
(\cite{Richter1975}).
\bigbreak
\textbf{\textcolor{orange}{Note:}} This document should probably not
start listing large cardinals, because \textbf{(0)} the fact that one
implies the other nonwithstanding, this is about “ordinals”, not
“cardinals”, \textbf{(1)} they are already well covered elsewhere
(see, e.g., \cite{Kanamori1997}) and \textbf{(2)} we don't want to
start making assumptions, e.g., about whether $\omega_1^L$ is or is
not equal to $\omega_1$, but without making such assumptions it is no
longer possible to correctly order definitions. Perhaps a median way
would be to assume $V=L$ for ordering, forget about measurable
cardinals and whatnot, and still include inaccessibles, Mahlo, weakly
compact, etc.
%
%
%
\section{Various statements}
\begin{prop}\label{PlusPlusStableOrdinalIsSigmaOneOneReflecting}
If $\alpha$ is such that $L_\alpha \mathrel{\preceq_1}
L_{\alpha^{++}}$ (where $\alpha^+,\alpha^{++}$ are the two smallest
admissible ordinals $>\alpha$) then $\alpha$ is
$\Sigma^1_1$-reflecting. (Stated in \cite[theorem 6.4 on
p. 376]{Simpson1978}.)
\end{prop}
\begin{proof}
Assume $L_\alpha \models \exists U(\varphi(U))$ where $\varphi$ is a
$\Pi^1_0$ (=first-order) formula with constants in $L_\alpha$ and the
extra relation symbol $U$. We want to show that there exists
$\beta<\alpha$ such that $L_\beta \models \exists U(\varphi(U))$.
Now by \cite[theorem 6.2 on p. 334]{RichterAczel1974} (applied to the
negation of $\exists U(\varphi(U))$) we can find a $\Pi_1$ formula
$\forall z(\psi(S,z))$ (with the same constants as $\varphi$) such
that for any countable transitive set $A$ containing these constants
and any admissible $B\ni A$ we have $B \models \forall z(\theta(A,z))$
iff $A \models \exists U(\varphi(U))$.
In particular, $L_{\alpha^+} \models \forall z(\theta(L_\alpha,z))$.
So $L_{\alpha^+} \models \exists A(\trans(A) \land \penalty0
(A\models\Theta+V{=}L) \land \penalty0 \forall z(\theta(A,z)))$, were
$\Theta$ is a statement which translates the adequacy of $A$ (see
\cite{Jech1978} (13.9) and lemmas 13.2 and 13.3 p. 110–112, or
\cite{Jech2003}, (13.12) and (13.13) p. 188). So in turn
$L_{\alpha^{++}} \models \exists C(\trans(C) \land \penalty0
(C\models\mathsf{KP}+V{=}L) \land \penalty0 (C \models \exists
A(\trans(A) \land \penalty100 (A\models\Theta+V{=}L) \land \penalty100
\forall z(\theta(A,z)))))$. But this is a $\Sigma_1$ formula with
constants in $L_\alpha$, so by the assumption we have $L_\alpha
\models$ the same thing. So there is $C \in L_\alpha$ transitive and
containing the constants of $\varphi$, and which is necessarily an
$L_\gamma$ (for $\gamma<\alpha$) because $C \models
\mathsf{KP}+V{=}L$, such that $L_\gamma \models \exists A(\trans(A)
\land \penalty0 (A\models\Theta+V{=}L) \land \penalty0 \forall
z(\theta(A,z)))$. So in turn there exists $A \in L_\gamma$
transitive, which is necessarily an $L_\beta$ (for $\beta<\gamma$)
because $A \models \Theta+V{=}L$, such that $L_\gamma \models \forall
z(\theta(L_\beta,z))$. So $L_\beta \models \exists U(\varphi(U))$.
\end{proof}
\begin{prop}\label{KPExistenceOfOmegaOneImpliesExistenceOfPOmega}
The following holds in $\mathsf{KP}$: if $A\subseteq \omega$ is
constructible, then $A \in L_\gamma$ for some countable
ordinal $\gamma$.
In particular, in $\mathsf{KP} + V=L$, if there exists an uncountable
ordinal $\delta$, then $\mathscr{P}(\omega)$ exists and can be defined
as $\{A \in L_\delta : A\subseteq\omega\}$.
\end{prop}
\begin{proof}
We have to verify that the usual proof (cf. \cite[chapter II,
lemma 5.5 on p. 84]{Devlin1984} or \cite[lemma 13.1 on
p. 110]{Jech1978} or \cite[theorem 13.20 on p. 190]{Jech2003})
works in $\mathsf{KP}$.
Working in $L$, we can assume that $V=L$ holds. Also, we can assume
that $\omega$ exists because if every set is finite the result is
trivial.
Since $A$ is constructible there is $\delta$ limit such that $A \in
L_\delta$. We can define $\Delta_1$-Skolem functions for $L_\delta$
inside $\mathsf{KP}$, and because $\omega$ exists we can use induction
(cf. \cite[remarks following definition 9.1 on p. 38]{Barwise1975}) to
construct the Skolem hull $M$ of $L_\omega \cup \{A\}$ inside
$L_\delta$ (or use \cite[chapter II, lemma 5.3 on p. 83]{Devlin1984}).
Since $M$ is extensional, we can now use the Mostowski collapse $\pi
\colon M \buildrel\sim\over\to N$ (cf. \cite[theorem 7.4 on
p. 32]{Barwise1975}) to collapse $M$ to a transitive set $N$, which
is necessarily an $L_\gamma$. Now $M$ is countable by construction,
so $N = L_\gamma$ is also, so $\gamma$ is. And we have $\pi(A) = A$
so $A \in L_\gamma$ with $\gamma$ countable, as asserted.
\end{proof}
%
%
%
\begin{thebibliography}{}
\bibitem[Aanderaa1974]{Aanderaa1974} Stål Aanderaa, “Inductive
Definitions and their Closure Ordinals”, \textit{in}: Jens Erik
Fenstad \& Peter G. Hinman (eds.), \textit{Generalized Recursion
Theory} (Oslo, 1972), North-Holland (1974), ISBN 0-7204-2276-0,
207–220.
\bibitem[AbramsonSacks1976]{AbramsonSacks1976} Fred G. Abramson \&
Gerald E. Sacks, “Uncountable Gandy Ordinals”, \textit{J. London
Math. Soc. (2)} \textbf{14} (1976), 387–392.
\bibitem[Aczel1970]{Aczel1970} Peter Aczel, “Representability in Some
Systems of Second Order Arithmetic”, \textit{Israel J. Math}
\textbf{8} (1970), 308–328.
\bibitem[AczelHinman1974]{AczelHinman1974} Peter Aczel \& Peter
G. Hinman, “Recursion in the Superjump”, \textit{in}: Jens Erik
Fenstad \& Peter G. Hinman (eds.), \textit{Generalized Recursion
Theory} (Oslo, 1972), North-Holland (1974), ISBN 0-7204-2276-0,
5–41.
\bibitem[Adams1981]{Adams1981} Douglas Adams, \textit{The Hitchiker's
Guide to the Galaxy}, Pocket Books (1981), ISBN 0-671-46149-4.
\bibitem[Avigad2002]{Avigad2002} Jeremy Avigad, “An ordinal analysis
of admissible set theory using recursion on ordinal notations”,
\textit{J. Math. Log.} \textbf{2} (2002), 91–112.
\bibitem[Barwise1975]{Barwise1975} Jon Barwise, \textit{Admissible
sets and structures, An approach to definability theory},
Perspectives in Mathematical Logic \textbf{7}, Springer-Verlag
(1975), ISBN 3-540-07451-1.
\bibitem[Cenzer1974]{Cenzer1974} Douglas Cenzer, “Ordinal Recursion
and Inductive Definitions”, \textit{in}: Jens Erik Fenstad \& Peter
G. Hinman (eds.), \textit{Generalized Recursion Theory} (Oslo,
1972), North-Holland (1974), ISBN 0-7204-2276-0, 221–264.
\bibitem[Devlin1984]{Devlin1984} Keith Devlin,
\textit{Constructibility}, Perspectives in Mathematical
Logic \textbf{6}, Springer-Verlag (1984), ISBN 3-540-13258-9.
\bibitem[Gostanian1979]{Gostanian1979} Richard Gostanian “The next
admissible ordinal”, \textit{Ann. Math. Logic} \textbf{17} (1979),
171–203.
\bibitem[GostanianHrbáček1979]{GostanianHrbacek1979} Richard Gostanian
\& Karel Hrbáček, “A new proof that $\pi^1_1 < \sigma^1_1$”,
\textit{Z. Math. Logik Grundlag. Math.} \textbf{25} (1979), 407–408.
\bibitem[Harrington1974]{Harrington1974} Leo Harrington, “The
Superjump and the first Recursively Mahlo Ordinal”, \textit{in}:
Jens Erik Fenstad \& Peter G. Hinman (eds.), \textit{Generalized
Recursion Theory} (Oslo, 1972), North-Holland (1974),
ISBN 0-7204-2276-0, 43–52.
\bibitem[Harrington1975]{Harrington1975} Leo Harrington, “Kolmogorov's
$R$-operator and the first nonprojectible ordinal”, unpublished
notes (1975).
\bibitem[Harrison1968]{Harrison1968} Joseph Harrison, “Recursive
pseudo-well-orderings”, \textit{Trans. Amer. Math. Soc.}
\textbf{131} (1968), 526–543.
\bibitem[Hinman1978]{Hinman1978} Peter G. Hinman,
\textit{Recursion-Theoretic Hierarchies}, Perspectives in
Mathematical Logic \textbf{9}, Springer-Verlag (1978),
ISBN 3-540-07904-1.
\bibitem[HinmanMoschovakis1971]{HinmanMoschovakis1971} Peter G. Hinman
\& Yiannis N. Moschovakis, “Computability over the Continuum”,
\textit{in}: R. O. Gandy \& C. M. E. Yates (eds.), \textit{Logic
Colloquium '69} (Manchester, 1969), North-Holland (1971), 77–105.
\bibitem[JaegerPohlers1983]{JaegerPohlers1983} Gerhard Jäger \&
Wolfram Pohlers, “Eine beweistheoretische Untersuchung von
($\Delta^1_2$-$\mathsf{CA}$)+($\mathsf{BI}$) und verwandter
Systeme”, \textit{Bayer. Akad. Wiss.,
Math.-Natur. Kl. Sitzungsber. 1982} (1983), 1–28.
\bibitem[Jech1978]{Jech1978} Thomas Jech, \textit{Set theory}, Pure
and Applied Mathematics \textbf{79}, Academic Press (1978),
ISBN 0-12-381950-4.
\bibitem[Jech2003]{Jech2003} Thomas Jech, \textit{Set theory, The
third millennium edition, revised and expanded}, Springer Monographs
in Mathematics, Springer-Verlag (2003), ISBN 3-540-44085-2.
\bibitem[Jensen1972]{Jensen1972} Ronald Björn Jensen, “The fine
structure of the constructible hierarchy”, \textit{Ann. Math. Logic}
\textbf{4} (1972), 229–308.
\bibitem[John1986]{John1986} Thomas John, “Recursion in Kolmogorov's
$R$-operator and the ordinal $\sigma_3$”, \textit{J. Symbolic Logic}
\textbf{51} (1986), 1–11.
\bibitem[Kanamori1997]{Kanamori1997} Akihiro Kanamori, \textit{The
Higher Infinite} (corrected first edition), Perspectives in
Mathematical Logic, Springer-Verlag (1997), ISBN 3-540-57071-3.
\bibitem[MarekSrebrny1973]{MarekSrebrny1973} Wiktor Marek \& Marian
Srebrny, “Gaps in the Constructible Universe”,
\textit{Ann. Math. Logic} \textbf{6} (1974), 359–394.
\bibitem[Nadel1973]{Nadel1973} Mark Nadel, “Scott Sentences and
Admissible Sets”, \textit{Ann. Math. Logic} \textbf{7} (1974),
267–294.
\bibitem[Putnam1963]{Putnam1963} Hilary Putnam, “A Note on
Constructible Sets of Integers”, \textit{Notre Dame J. Formal Logic}
\textbf{4} (1963), 270–273.
\bibitem[Rathjen1990]{Rathjen1990} Michael Rathjen, “Ordinal Notations
Based on a Weakly Mahlo Cardinal”, \textit{Arch. Math. Logic}
\textbf{29} (1990), 249–263.
\bibitem[Rathjen1994]{Rathjen1994} Michael Rathjen, “Proof theory of
reflection”, \textit{Ann. Pure Appl. Logic} \textbf{68} (1994),
181–224.
\bibitem[Richter1975]{Richter1975} Wayne Richter, “The Least
$\Sigma^1_2$ and $\Pi^1_2$ Reflecting Ordinals”, \textit{in}: Gert
H. Müller, Arnold Oberschelp \& Klaus Potthoff,
\textit{$\models$ISILC Logic Conference} (Kiel, 1974),
Springer-Verlag \textit{Lecture Notes in Math.} \textbf{499} (1975),
ISBN 3-540-07534-8, 568–578.
\bibitem[RichterAczel1974]{RichterAczel1974} Wayne Richter \& Peter
Aczel, “Inductive Definitions and Reflecting Properties of
Admissible Ordinals”, \textit{in}: Jens Erik Fenstad \& Peter
G. Hinman (eds.), \textit{Generalized Recursion Theory} (Oslo,
1972), North-Holland (1974), ISBN 0-7204-2276-0, 301–381.
\bibitem[Simpson1978]{Simpson1978} Stephen G. Simpson, “Short Course
on Admissible Recursion Theory”, \textit{in}: Jens Erik Fenstad,
R. O. Gandy \& Gerald E. Sacks (eds.), \textit{Generalized Recursion
Theory II} (Oslo, 1977), North-Holland (1978), ISBN 0-444-85163-1,
355–390.
\bibitem[Simpson2009]{Simpson2009} Stephen G. Simpson,
\textit{Subsystems of Second-Order Arithmetic} (second edition),
Perspectives in Logic, ASL (2009), ISBN 978-0-521-88439-6.
\bibitem[Stegert2010]{Stegert2010} Jan-Carl Stegert, \textit{Ordinal
Proof Theory of Kripke-Platek Set Theory Augmented by Strong
Reflection Principles}, PhD dissertation (Westfälischen
Wilhelms-Universität Münster), 2010.
\end{thebibliography}
\end{document}